Simple Subsequence Problem

You are given a set S of strings consisting of 0 and 1, and an integer K.

Find the longest string that is a subsequence of K or more different strings in S. If there are multiple strings that satisfy this condition, find the lexicographically smallest such string.

Here, S is given in the format below:

• The data directly given to you is an integer N, and N+1 strings X_0,X_1,...,X_N. For every i (0\leq i\leq N), the length of X_i is 2^i.
• For every pair of two integers (i,j) (0\leq i\leq N,0\leq j\leq 2^i-1), the j-th character of X_i is 1 if and only if the binary representation of j with i digits (possibly with leading zeros) belongs to S. Here, the first and last characters in X_i are called the 0-th and (2^i-1)-th characters, respectively.
• S does not contain a string with length N+1 or more.

Here, a string A is a subsequence of another string B when there exists a sequence of integers t_1 < ... < t_{|A|} such that, for every i (1\leq i\leq |A|), the i-th character of A and the t_i-th character of B is equal.

Constraints

• 0 \leq N \leq 20
• X_i(0\leq i\leq N) is a string of length 2^i consisting of 0 and 1.
• 1 \leq K \leq |S|
• K is an integer.

Input

Input is given from Standard Input in the following format:

N K X_0 : X_N

Output

Print the lexicographically smallest string among the longest strings that are subsequences of K or more different strings in S.

Examples

Input

3 4 1 01 1011 01001110

Output

10

Input

4 6 1 01 1011 10111010 1101110011111101

Output

100

Input

2 5 0 11 1111

Output

inputFormat

Input

Input is given from Standard Input in the following format:

N K X_0 : X_N

outputFormat

Output

Print the lexicographically smallest string among the longest strings that are subsequences of K or more different strings in S.

Examples

Input

3 4 1 01 1011 01001110

Output

10

Input

4 6 1 01 1011 10111010 1101110011111101

Output

100

Input

2 5 0 11 1111

Output

样例

4 6
1
01
1011
10111010
1101110011111101

100